If there were a black hole orbiting an inhabited planet, with the gravity of a typical moon (e.g our moon, orbiting a similarly weighted planet to ours), what practical differences would arise when compared to a normal moon (such as it affecting the atmosphere) and would there be a viable way to harvest energy from it?
what practical differences would arise when compared to a normal moon (such as it affecting the atmosphere)
There'd be no moonlight under normal circumstances, which means that no life would have evolved to make use of that, and you wouldn't find any moon-rocks on Earth as there would never be any impacts as such.
There'd be some hazards associated with an accretion disk forming, because those can get pretty hot. If the planet has an atmosphere though, you can reasonably assume that the accretion disk is either absent or fairly small and benign.
and would there be a viable way to harvest energy from it?
Not trivially, though you can do tricks with rotating black holes that have what is called an ergosphere, where if you enter the ergosphere travelling fast enough then you can steal a little bit of the angular momentum of the spinning black hole and leave a little faster than when you went in. Extracting energy from this is tricky, but using it as a boost for spacecraft should be a bit easier.
Because the radius of the black hole is so small, its rotation would not have any additional tidal effects on Earth, and it would not have undergone tidal locking in the way that the real moon did.
Just to note, the evaporation timescale of black holes is long. Even a hole as small a billion tonnes will take over a trillion years to evaporate, and the moon is a hundreds of million times heavier than that. The relevant wikipedia page has the full gory details, though the important thing is that evaporation time scales with the cube of the mass of the black hole.
Don't worry about your local black hole exploding and destroying your planet... think more about the fun fireworks when the local sun finally enters the red giant phase of its life after a few billion years, and its photosphere expands to cover the orbit of your planet...
There is, almost but not quite inevitably, an XKCD about that (or at least, a what-if). It doesn't consider extracting energy, but the rest may be of interest. To summarise: looks boring, has a near-undetectable reduction in the heating of the Earth.
would there be a viable way to harvest energy from it?
Since gravitationally this is almost identical with our situation, there is a way of harvesting some energy - the black hole would produce tides of the same amplitude we are used to. And that can be used to generate electricity...
If our Moon were swapped by a black hole of equal mass, there would be no gravitational distinction felt on Earth. We would miss out on Lunar eclipses and solar eclipses would probably make the Sun look like a ring while in "totality."
Creatures that use the light of the Moon phases would get messed up and may not breed (theoretically, this is when corals know when to breed).
It would be pretty dark at night too.
Throwing stuff at the Moon massed black hole would release energy as the matter is ripped apart before it crosses the event horizon. However, to collect that energy, you would want to surround the Moon-hole at a "safe" distance. I don't know if that would be worth it.
Another problem is that a black hole that low in mass would be relatively short lived. Black holes evaporate. A Moon sized back hole wouldn't capture enough matter to offset the evaporation. Eventually, it would decrease in mass enough to not be able to hold itself together and go boom. That would be a bad day.
I would say that it is unlikely to be found naturally existing. Since any low mass black holes created in the big bang would have already evaporated away.
That means that super science would be needed to create one. I think that any science that could create a Moon massed black hole wouldn't need the energy generated one.
Others have already answered the boring part of this question (= very low impact on earth as we know it, apart from the uses that biology and human culture make of the moonlight). I won't repeat that. I'm going to answer the fun part of the question: How to harvest energy from it.
First of all, a moon-mass BH is tiny. Wikipedia puts it at 0.109 millimeters. That's big enough to interact with macroscopic objects, but tiny compared to human made space hardware. As such, I see two distinct ways of harvesting meaningful amounts of energy from it:
Option 1: The spacecraft slinger
Build a long (= tens to hundreds of kilometers) beam with loads of tensile strength.
Send it into orbit around the BH where the tidal forces on it are still manageable. This orbit will have a radius that is also in the range of tens to hundreds of kilometers. Put it into rotation such that the same end always points towards the BH.
Have a docking station for spacecraft at the point where the orbital speed of the beam structure matches the escape velocity.
Have two escalators starting at that point, one going outwards for the spacecraft, the other one going inwards for small weights.
Each time a spacecraft docks at the spacecraft slinger, it brings a matching weight with it. The weight is moved inwards to the very end of the beam, the spacecraft is moved outwards far enough that the orbit of the entire structure remains at the same height above the BH. Then both, the weight and the spacecraft are released at the right moment to send the spacecraft on the intended trajectory.
This is essentially a free floating space elevator that operates in a much tighter gravity well than earth's, and that operates at a much higher angular speed than earth's rotation. These two features make the design much more easy to build. However, the weights are required because the massive rotating body at the center of the gravity well is missing.
Since the beam reaches quite far down into the potential well of the BH, the weight can be tiny compared to the spacecraft.
As an added bonus, both the weight and the spacecraft lose potential energy as they are lowered to their respective release points (well the weight first needs to be elevated against the centrifugal force, but that energy is more than regained as the weight moves further into the gravity well). As such, the operation of the spacecraft slinger actually produces energy.
Obviously, the weights will accumulate in a belt of highly elliptical orbits around the BH below the lower end of the elevator. In the long run, this will enclose the BH into a cloud of Kessler debris, and that cloud will gradually heat up as collisions become more and more frequent. This won't pose a big threat to the elevator though, because a) the collisions will tend to circularize the orbits of the fragments, b) considerable energy is lost by grinding down the fragments, and c) the weights only barely had the energy to touch the lower end of the elevator to begin with. However, this Kessler cloud will make the realization of option 2 much harder.
Option 2: The Dyson Power Plant
Build a spherical megastructure around the BH.
Line the inside of that megastructure with generators that can turn hard EM radiation into electricity.
Add a device that is able to aim millimeter sized bullets precisely at the BH.
Add some doors to the sphere that can be opened to provide a tiny propulsion to the sphere to keep it centered exactly on the BH.
It does not really matter, what those bullets are made out of. Only the denser, the better. Each bullet will accelerate to relativistic speeds as it falls towards the BH. When it hits the BH, most of the bullet's mass will move right past it. However, all of the bullets mass will be deflected towards the BH, and will subsequently crash into itself (the bullet will actually behave like a giant cluster of ping pong balls, not like a rigid body; the energies involved are much too high for inter-atomic forces to play a significant role). This will instantly turn the bullet into an extremely hot plasma that emits loads of gamma rays, allowing the surrounding sphere's generators to do their thing. The plasma won't retain enough kinetic energy to get back to the Dyson sphere, it will fall back onto the BH until it turns into a tiny accretion disk that continues to produce hard EM radiation.
It would be possible to run this gigantic power plant without entirely enclosing the BH in a Dyson sphere, but that would have two problems: 1) Most of the produced energy would be lost to space, and 2) if there are people living on a planet nearby, they might not be too happy with the effects the intense gamma radiation has on their atmosphere. Or with the effects that the UV light has on their eyes. It would be like a continuous atomic explosion in space. Of course, it would be possible to build a Dyson ring that ensures that the BH is never visible from inhabited planets. Nevertheless, astronauts in spaceships might not be appeased by that.
I just want to add that on top of all the other answers explaining the tricks of light and how to extract energy, eventually civilizations in the planet will figure out the existence of the black hole, and quite early too (around Isaac Newton's era, if they have a similar history to ours).
They might send rockets there during their early space age with potentially disastrous results. You see, the gravitational effects ON EARTH are indistinct from an actual moon, but close to the hole is something else entirely.
The gravity pull of a body decreases with distance from its center of mass. But for planets, if you dig into their mantles, there is a shell effect that reduces gravity because there is less mass below you and more mass pulling you up. The result being that the largest pull you get is when standing on their surface. That said, if the radius of your "moon" is smaller than your hand...
Remember that gravity becomes 4x stronger every time you reduce your distance to the source by half. The radius of the moon is just a little less than 231 greater than the event horizon of the black hole in the question (I may be wrong by one or two orders of magnitude, but at this level it doesn't matter much). So if a rocket passes even a few meters close to hole, the rocket will be torn apart and its particles will be accelerated to near light speeds.
On the bright side, scientists can use the black hole to achieve awesome slingshot maneuvers. Just aim to a few hundreds of kilometers away from the black hole and it will be like having Jupiter on our backyard. They could send missions to outer planets and beyond much sooner and at much less cost.
On the dark side, if you make a mistake and the rocket passes too close to the black hole, the rocket's disassembled molecular or even subatomic parts might be coming back towards the Earth at relativistic speed, causing more damage than huge meteor strikes. Our very first mission to investigate the black hole could also be our last.
By the way, that is considering plot armor. It might be that asteroids such as the one that did the dinosaurs could take a gravity slingshot from this black hole too, hitting the Earth at thousands of times faster speeds than they would otherwise. This has the potential to vaporize our planet. Since there are literal geological eras of heavy asteroid and comet bombardment on Earth and on the Moon (consider all her craters), it is more likely that the Earth won't even exist unless the black hole has appeared relatively recent.
The radius of a black hole is related to its mass by the simple formula R = 3*M, where the mass M is given in units of the Sun's mass and the radius R is the radius of the black hole's event horizon given in kilometers.
The mass of the Moon is about
7.34 * 10^22 kilograms
The mass of the Sun is about
1.99 * 10^30 kilograms
The mass of the Moon is therefore about
4 * 10^-8 in Sol equivalents
The black hole therefore has an event horizon radius of about
12 * 10^-8 kilometers
If I did the arithmetic correctly, this is about 0.012 millimeters. For perspective, it's a bit less than the size of a human hair. An object of this size probably would not result in any detectable disturbance in the light reaching earth.
Anything that came within that (very small) radius would be unable to escape the gravitational well, it's true. But I think we could spare the odd bit of interstellar dust. :)
If the Moon were replaced by such a black hole, the gravitational effects you experience would be unchanged so long as you remain at a distance greater than or equal to the original radius of the Moon. However, since the gravitational force experienced depends on the distance between the center of masses and is given by
F = G M1 M2 / D^2, the gravitational force (and therefore the gravitational acceleration) experienced will be greater as you move in closer than the original radius of the Moon.
Since current rocket designs carry only barely enough fuel to achieve escape velocity starting at the surface with one earth-normal gravity (9.8 meters/sec^2), it is interesting to calculate at what distance from the microMoon you would experience that same gravitational acceleration.
The things you might do to harness those effects here on earth would be the same things we can do with the Moon's gravitational effects. Up close and personal might be a bit different...
A few points that haven't been raised yet:
The moon-sized black hole would shelter Earth ~negligibly from impacts (e.g., Late Heavy Bombardment) probably increasing the duty cycle of extinction-level events, possibly accelerating evolution but also shortening the timespan for intelligent life to become space-faring and escape oblivion.
The same ~centimeter sized debris from the tails of comets that produce our meteor showers could be gravitationally slingshotted from the black hole towards the Earth with potentially catastrophic effects. This could be expanded upon.
If the black hole's spin axis aligns with the Earth, we'd be safe. However, if the spin axis were to point to Earth (occasionally) then in the case of periods of accretion (e.g., tidal disruption events of asteroids etc) the Earth would be flooded with gamma rays from the black hole's jet.
Supposing life somehow develops like on the present Earth and becomes ~space-faring, we'd have a laboratory to test and develop quantum theories of gravity, vastly accelerating scientific progress.